In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of. Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise.

Statement

If is a zero-mean multivariate normal random vector, thenwhere the sum is over all the pairings of, i.e. all distinct ways of partitioning into pairs {i,j}, and the product is over the pairs contained in p. More generally, if is a zero-mean complex-valued multivariate normal random vector, then the formula still holds. The expression on the right-hand side is also known as the hafnian of the covariance matrix of.

Odd case

If n=2m+1 is odd, there does not exist any pairing of. Under this hypothesis, Isserlis' theorem implies that This also follows from the fact that has the same distribution as X, which implies that.

Even case

In his original paper, Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the order moments, which takes the appearance If n=2m is even, there exist (see double factorial) pair partitions of : this yields terms in the sum. For example, for order moments (i.e. 4 random variables) there are three terms. For -order moments there are terms, and for -order moments there are terms.

Example

We can evaluate the characteristic function of gaussians by the Isserlis theorem:

Proof

Since both sides of the formula are multilinear in, if we can prove the real case, we get the complex case for free. Let be the covariance matrix, so that we have the zero-mean multivariate normal random vector. Since both sides of the formula are continuous with respect to \Sigma, it suffices to prove the case when \Sigma is invertible. Using quadratic factorization, we get Differentiate under the integral sign with to obtain . That is, we need only find the coefficient of term in the Taylor expansion of. If n is odd, this is zero. So let n = 2m, then we need only find the coefficient of term in the polynomial. Expand the polynomial and count, we obtain the formula. \square

Generalizations

Gaussian integration by parts

An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts. If is a zero-mean multivariate normal random vector, then This is a generalization of Stein's lemma. The Wick's probability formula can be recovered by induction, considering the function defined by. Among other things, this formulation is important in Liouville conformal field theory to obtain conformal Ward identities, BPZ equations and to prove the Fyodorov-Bouchaud formula.

Non-Gaussian random variables

For non-Gaussian random variables, the moment-cumulants formula replaces the Wick's probability formula. If is a vector of random variables, then where the sum is over all the partitions of, the product is over the blocks of p and is the joint cumulant of.